Properties

Label 390.2.a.f.1.1
Level $390$
Weight $2$
Character 390.1
Self dual yes
Analytic conductor $3.114$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 390.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +4.00000 q^{22} +8.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +6.00000 q^{29} -1.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -10.0000 q^{37} +4.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} +4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} +8.00000 q^{46} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} +1.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +4.00000 q^{55} -4.00000 q^{57} +6.00000 q^{58} +4.00000 q^{59} -1.00000 q^{60} -2.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -4.00000 q^{66} -12.0000 q^{67} -6.00000 q^{68} -8.00000 q^{69} +16.0000 q^{71} +1.00000 q^{72} +2.00000 q^{73} -10.0000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -1.00000 q^{78} -16.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} -6.00000 q^{85} +4.00000 q^{86} -6.00000 q^{87} +4.00000 q^{88} +10.0000 q^{89} +1.00000 q^{90} +8.00000 q^{92} +8.00000 q^{93} +4.00000 q^{95} -1.00000 q^{96} -6.00000 q^{97} -7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −1.00000 −0.182574
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 4.00000 0.648886
\(39\) −1.00000 −0.160128
\(40\) 1.00000 0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 4.00000 0.603023
\(45\) 1.00000 0.149071
\(46\) 8.00000 1.17954
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) 6.00000 0.840168
\(52\) 1.00000 0.138675
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 6.00000 0.787839
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) −4.00000 −0.492366
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −6.00000 −0.727607
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −10.0000 −1.16248
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 4.00000 0.431331
\(87\) −6.00000 −0.643268
\(88\) 4.00000 0.426401
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) −1.00000 −0.102062
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −7.00000 −0.707107
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 6.00000 0.594089
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 4.00000 0.381385
\(111\) 10.0000 0.949158
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) −4.00000 −0.374634
\(115\) 8.00000 0.746004
\(116\) 6.00000 0.557086
\(117\) 1.00000 0.0924500
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 6.00000 0.541002
\(124\) −8.00000 −0.718421
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 1.00000 0.0877058
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) −1.00000 −0.0860663
\(136\) −6.00000 −0.514496
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −8.00000 −0.681005
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 16.0000 1.34269
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) 2.00000 0.165521
\(147\) 7.00000 0.577350
\(148\) −10.0000 −0.821995
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 4.00000 0.324443
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) −1.00000 −0.0800641
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −16.0000 −1.27289
\(159\) 10.0000 0.793052
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −6.00000 −0.468521
\(165\) −4.00000 −0.311400
\(166\) −12.0000 −0.931381
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −6.00000 −0.460179
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) −4.00000 −0.300658
\(178\) 10.0000 0.749532
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 1.00000 0.0745356
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 8.00000 0.589768
\(185\) −10.0000 −0.735215
\(186\) 8.00000 0.586588
\(187\) −24.0000 −1.75505
\(188\) 0 0
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −6.00000 −0.430775
\(195\) −1.00000 −0.0716115
\(196\) −7.00000 −0.500000
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 4.00000 0.284268
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 1.00000 0.0707107
\(201\) 12.0000 0.846415
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) 8.00000 0.556038
\(208\) 1.00000 0.0693375
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −10.0000 −0.686803
\(213\) −16.0000 −1.09630
\(214\) −12.0000 −0.820303
\(215\) 4.00000 0.272798
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) −2.00000 −0.135147
\(220\) 4.00000 0.269680
\(221\) −6.00000 −0.403604
\(222\) 10.0000 0.671156
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 10.0000 0.665190
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) −4.00000 −0.264906
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) −7.00000 −0.447214
\(246\) 6.00000 0.382546
\(247\) 4.00000 0.254514
\(248\) −8.00000 −0.508001
\(249\) 12.0000 0.760469
\(250\) 1.00000 0.0632456
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) 0 0
\(253\) 32.0000 2.01182
\(254\) 8.00000 0.501965
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) 1.00000 0.0620174
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) −4.00000 −0.246183
\(265\) −10.0000 −0.614295
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) −12.0000 −0.733017
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 4.00000 0.241209
\(276\) −8.00000 −0.481543
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 12.0000 0.719712
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 16.0000 0.949425
\(285\) −4.00000 −0.236940
\(286\) 4.00000 0.236525
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 6.00000 0.352332
\(291\) 6.00000 0.351726
\(292\) 2.00000 0.117041
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 7.00000 0.408248
\(295\) 4.00000 0.232889
\(296\) −10.0000 −0.581238
\(297\) −4.00000 −0.232104
\(298\) 6.00000 0.347571
\(299\) 8.00000 0.462652
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 0 0
\(303\) 2.00000 0.114897
\(304\) 4.00000 0.229416
\(305\) −2.00000 −0.114520
\(306\) −6.00000 −0.342997
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 10.0000 0.560772
\(319\) 24.0000 1.34374
\(320\) 1.00000 0.0559017
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) 1.00000 0.0554700
\(326\) 4.00000 0.221540
\(327\) 10.0000 0.553001
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) −4.00000 −0.220193
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −12.0000 −0.658586
\(333\) −10.0000 −0.547997
\(334\) 8.00000 0.437741
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 1.00000 0.0543928
\(339\) −10.0000 −0.543125
\(340\) −6.00000 −0.325396
\(341\) −32.0000 −1.73290
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) −8.00000 −0.430706
\(346\) −18.0000 −0.967686
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) −6.00000 −0.321634
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 4.00000 0.213201
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) −4.00000 −0.212598
\(355\) 16.0000 0.849192
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) 6.00000 0.315353
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 2.00000 0.104542
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 8.00000 0.417029
\(369\) −6.00000 −0.312348
\(370\) −10.0000 −0.519875
\(371\) 0 0
\(372\) 8.00000 0.414781
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −24.0000 −1.24101
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 4.00000 0.205196
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) 32.0000 1.63512 0.817562 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 4.00000 0.203331
\(388\) −6.00000 −0.304604
\(389\) −34.0000 −1.72387 −0.861934 0.507020i \(-0.830747\pi\)
−0.861934 + 0.507020i \(0.830747\pi\)
\(390\) −1.00000 −0.0506370
\(391\) −48.0000 −2.42746
\(392\) −7.00000 −0.353553
\(393\) −12.0000 −0.605320
\(394\) 6.00000 0.302276
\(395\) −16.0000 −0.805047
\(396\) 4.00000 0.201008
\(397\) 30.0000 1.50566 0.752828 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 12.0000 0.598506
\(403\) −8.00000 −0.398508
\(404\) −2.00000 −0.0995037
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −40.0000 −1.98273
\(408\) 6.00000 0.297044
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) −6.00000 −0.296319
\(411\) −10.0000 −0.493264
\(412\) 0 0
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) −12.0000 −0.589057
\(416\) 1.00000 0.0490290
\(417\) −12.0000 −0.587643
\(418\) 16.0000 0.782586
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) −12.0000 −0.584151
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) −6.00000 −0.291043
\(426\) −16.0000 −0.775203
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) −4.00000 −0.193122
\(430\) 4.00000 0.192897
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 0 0
\(435\) −6.00000 −0.287678
\(436\) −10.0000 −0.478913
\(437\) 32.0000 1.53077
\(438\) −2.00000 −0.0955637
\(439\) −40.0000 −1.90910 −0.954548 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) 4.00000 0.190693
\(441\) −7.00000 −0.333333
\(442\) −6.00000 −0.285391
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 10.0000 0.474579
\(445\) 10.0000 0.474045
\(446\) 8.00000 0.378811
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 1.00000 0.0471405
\(451\) −24.0000 −1.13012
\(452\) 10.0000 0.470360
\(453\) 0 0
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 34.0000 1.59045 0.795226 0.606313i \(-0.207352\pi\)
0.795226 + 0.606313i \(0.207352\pi\)
\(458\) 14.0000 0.654177
\(459\) 6.00000 0.280056
\(460\) 8.00000 0.373002
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 6.00000 0.278543
\(465\) 8.00000 0.370991
\(466\) −14.0000 −0.648537
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 4.00000 0.184115
\(473\) 16.0000 0.735681
\(474\) 16.0000 0.734904
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) −8.00000 −0.365911
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −10.0000 −0.455961
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −6.00000 −0.272446
\(486\) −1.00000 −0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −4.00000 −0.180886
\(490\) −7.00000 −0.316228
\(491\) 4.00000 0.180517 0.0902587 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) 6.00000 0.270501
\(493\) −36.0000 −1.62136
\(494\) 4.00000 0.179969
\(495\) 4.00000 0.179787
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 1.00000 0.0447214
\(501\) −8.00000 −0.357414
\(502\) −28.0000 −1.24970
\(503\) 8.00000 0.356702 0.178351 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) 32.0000 1.42257
\(507\) −1.00000 −0.0444116
\(508\) 8.00000 0.354943
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 6.00000 0.265684
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 1.00000 0.0438529
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 6.00000 0.262613
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 48.0000 2.09091
\(528\) −4.00000 −0.174078
\(529\) 41.0000 1.78261
\(530\) −10.0000 −0.434372
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) −10.0000 −0.432742
\(535\) −12.0000 −0.518805
\(536\) −12.0000 −0.518321
\(537\) −12.0000 −0.517838
\(538\) −26.0000 −1.12094
\(539\) −28.0000 −1.20605
\(540\) −1.00000 −0.0430331
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 24.0000 1.03089
\(543\) −6.00000 −0.257485
\(544\) −6.00000 −0.257248
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 10.0000 0.427179
\(549\) −2.00000 −0.0853579
\(550\) 4.00000 0.170561
\(551\) 24.0000 1.02243
\(552\) −8.00000 −0.340503
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) 10.0000 0.424476
\(556\) 12.0000 0.508913
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −8.00000 −0.338667
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 10.0000 0.421825
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) 20.0000 0.840663
\(567\) 0 0
\(568\) 16.0000 0.671345
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) −4.00000 −0.167542
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) 8.00000 0.333623
\(576\) 1.00000 0.0416667
\(577\) 42.0000 1.74848 0.874241 0.485491i \(-0.161359\pi\)
0.874241 + 0.485491i \(0.161359\pi\)
\(578\) 19.0000 0.790296
\(579\) −10.0000 −0.415586
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) 6.00000 0.248708
\(583\) −40.0000 −1.65663
\(584\) 2.00000 0.0827606
\(585\) 1.00000 0.0413449
\(586\) −26.0000 −1.07405
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 7.00000 0.288675
\(589\) −32.0000 −1.31854
\(590\) 4.00000 0.164677
\(591\) −6.00000 −0.246807
\(592\) −10.0000 −0.410997
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 8.00000 0.327418
\(598\) 8.00000 0.327144
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 2.00000 0.0812444
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) −12.0000 −0.484281
\(615\) 6.00000 0.241943
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) −8.00000 −0.321288
\(621\) −8.00000 −0.321029
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 1.00000 0.0400000
\(626\) −22.0000 −0.879297
\(627\) −16.0000 −0.638978
\(628\) −2.00000 −0.0798087
\(629\) 60.0000 2.39236
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) −16.0000 −0.636446
\(633\) 12.0000 0.476957
\(634\) −2.00000 −0.0794301
\(635\) 8.00000 0.317470
\(636\) 10.0000 0.396526
\(637\) −7.00000 −0.277350
\(638\) 24.0000 0.950169
\(639\) 16.0000 0.632950
\(640\) 1.00000 0.0395285
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 12.0000 0.473602
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) −24.0000 −0.944267
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 1.00000 0.0392837
\(649\) 16.0000 0.628055
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) 10.0000 0.391031
\(655\) 12.0000 0.468879
\(656\) −6.00000 −0.234261
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) −4.00000 −0.155700
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −20.0000 −0.777322
\(663\) 6.00000 0.233021
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) 48.0000 1.85857
\(668\) 8.00000 0.309529
\(669\) −8.00000 −0.309298
\(670\) −12.0000 −0.463600
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 2.00000 0.0770371
\(675\) −1.00000 −0.0384900
\(676\) 1.00000 0.0384615
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) −10.0000 −0.384048
\(679\) 0 0
\(680\) −6.00000 −0.230089
\(681\) −4.00000 −0.153280
\(682\) −32.0000 −1.22534
\(683\) 28.0000 1.07139 0.535695 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(684\) 4.00000 0.152944
\(685\) 10.0000 0.382080
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) 4.00000 0.152499
\(689\) −10.0000 −0.380970
\(690\) −8.00000 −0.304555
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) 12.0000 0.455186
\(696\) −6.00000 −0.227429
\(697\) 36.0000 1.36360
\(698\) 6.00000 0.227103
\(699\) 14.0000 0.529529
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −40.0000 −1.50863
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) −50.0000 −1.87779 −0.938895 0.344204i \(-0.888149\pi\)
−0.938895 + 0.344204i \(0.888149\pi\)
\(710\) 16.0000 0.600469
\(711\) −16.0000 −0.600047
\(712\) 10.0000 0.374766
\(713\) −64.0000 −2.39682
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 12.0000 0.448461
\(717\) 8.00000 0.298765
\(718\) 16.0000 0.597115
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) −18.0000 −0.669427
\(724\) 6.00000 0.222988
\(725\) 6.00000 0.222834
\(726\) −5.00000 −0.185567
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 2.00000 0.0740233
\(731\) −24.0000 −0.887672
\(732\) 2.00000 0.0739221
\(733\) −50.0000 −1.84679 −0.923396 0.383849i \(-0.874598\pi\)
−0.923396 + 0.383849i \(0.874598\pi\)
\(734\) −24.0000 −0.885856
\(735\) 7.00000 0.258199
\(736\) 8.00000 0.294884
\(737\) −48.0000 −1.76810
\(738\) −6.00000 −0.220863
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −10.0000 −0.367607
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 8.00000 0.293294
\(745\) 6.00000 0.219823
\(746\) 22.0000 0.805477
\(747\) −12.0000 −0.439057
\(748\) −24.0000 −0.877527
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) 28.0000 1.02038
\(754\) 6.00000 0.218507
\(755\) 0 0
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) −4.00000 −0.145287
\(759\) −32.0000 −1.16153
\(760\) 4.00000 0.145095
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) −8.00000 −0.289809
\(763\) 0 0
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) 32.0000 1.15621
\(767\) 4.00000 0.144432
\(768\) −1.00000 −0.0360844
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 10.0000 0.359908
\(773\) 38.0000 1.36677 0.683383 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) 4.00000 0.143777
\(775\) −8.00000 −0.287368
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) −34.0000 −1.21896
\(779\) −24.0000 −0.859889
\(780\) −1.00000 −0.0358057
\(781\) 64.0000 2.29010
\(782\) −48.0000 −1.71648
\(783\) −6.00000 −0.214423
\(784\) −7.00000 −0.250000
\(785\) −2.00000 −0.0713831
\(786\) −12.0000 −0.428026
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 6.00000 0.213741
\(789\) −24.0000 −0.854423
\(790\) −16.0000 −0.569254
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) −2.00000 −0.0710221
\(794\) 30.0000 1.06466
\(795\) 10.0000 0.354663
\(796\) −8.00000 −0.283552
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 10.0000 0.353333
\(802\) 34.0000 1.20058
\(803\) 8.00000 0.282314
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 26.0000 0.915243
\(808\) −2.00000 −0.0703598
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 1.00000 0.0351364
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 0 0
\(813\) −24.0000 −0.841717
\(814\) −40.0000 −1.40200
\(815\) 4.00000 0.140114
\(816\) 6.00000 0.210042
\(817\) 16.0000 0.559769
\(818\) −22.0000 −0.769212
\(819\) 0 0
\(820\) −6.00000 −0.209529
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) −10.0000 −0.348790
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 8.00000 0.278019
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) −12.0000 −0.416526
\(831\) 10.0000 0.346896
\(832\) 1.00000 0.0346688
\(833\) 42.0000 1.45521
\(834\) −12.0000 −0.415526
\(835\) 8.00000 0.276851
\(836\) 16.0000 0.553372
\(837\) 8.00000 0.276520
\(838\) −4.00000 −0.138178
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −34.0000 −1.17172
\(843\) −10.0000 −0.344418
\(844\) −12.0000 −0.413057
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 0 0
\(848\) −10.0000 −0.343401
\(849\) −20.0000 −0.686398
\(850\) −6.00000 −0.205798
\(851\) −80.0000 −2.74236
\(852\) −16.0000 −0.548151
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) −12.0000 −0.410152
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) −4.00000 −0.136558
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −18.0000 −0.612018
\(866\) 34.0000 1.15537
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) −64.0000 −2.17105
\(870\) −6.00000 −0.203419
\(871\) −12.0000 −0.406604
\(872\) −10.0000 −0.338643
\(873\) −6.00000 −0.203069
\(874\) 32.0000 1.08242
\(875\) 0 0
\(876\) −2.00000 −0.0675737
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −40.0000 −1.34993
\(879\) 26.0000 0.876958
\(880\) 4.00000 0.134840
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) −7.00000 −0.235702
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) −6.00000 −0.201802
\(885\) −4.00000 −0.134459
\(886\) 36.0000 1.20944
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 10.0000 0.335578
\(889\) 0 0
\(890\) 10.0000 0.335201
\(891\) 4.00000 0.134005
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) −8.00000 −0.267112
\(898\) 18.0000 0.600668
\(899\) −48.0000 −1.60089
\(900\) 1.00000 0.0333333
\(901\) 60.0000 1.99889
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) 6.00000 0.199447
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 4.00000 0.132745
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −4.00000 −0.132453
\(913\) −48.0000 −1.58857
\(914\) 34.0000 1.12462
\(915\) 2.00000 0.0661180
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) 6.00000 0.198030
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 8.00000 0.263752
\(921\) 12.0000 0.395413
\(922\) −18.0000 −0.592798
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 8.00000 0.262896
\(927\) 0 0
\(928\) 6.00000 0.196960
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 8.00000 0.262330
\(931\) −28.0000 −0.917663
\(932\) −14.0000 −0.458585
\(933\) 24.0000 0.785725
\(934\) 12.0000 0.392652
\(935\) −24.0000 −0.784884
\(936\) 1.00000 0.0326860
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 22.0000 0.717943
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 2.00000 0.0651635
\(943\) −48.0000 −1.56310
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 16.0000 0.519656
\(949\) 2.00000 0.0649227
\(950\) 4.00000 0.129777
\(951\) 2.00000 0.0648544
\(952\) 0 0
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) −10.0000 −0.323762
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) −24.0000 −0.775810
\(958\) −8.00000 −0.258468
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) −10.0000 −0.322413
\(963\) −12.0000 −0.386695
\(964\) 18.0000 0.579741
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) 5.00000 0.160706
\(969\) 24.0000 0.770991
\(970\) −6.00000 −0.192648
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) −1.00000 −0.0320256
\(976\) −2.00000 −0.0640184
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) −4.00000 −0.127906
\(979\) 40.0000 1.27841
\(980\) −7.00000 −0.223607
\(981\) −10.0000 −0.319275
\(982\) 4.00000 0.127645
\(983\) 8.00000 0.255160 0.127580 0.991828i \(-0.459279\pi\)
0.127580 + 0.991828i \(0.459279\pi\)
\(984\) 6.00000 0.191273
\(985\) 6.00000 0.191176
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 32.0000 1.01754
\(990\) 4.00000 0.127128
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −8.00000 −0.254000
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) 12.0000 0.380235
\(997\) −58.0000 −1.83688 −0.918439 0.395562i \(-0.870550\pi\)
−0.918439 + 0.395562i \(0.870550\pi\)
\(998\) 20.0000 0.633089
\(999\) 10.0000 0.316386
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 390.2.a.f.1.1 1
3.2 odd 2 1170.2.a.a.1.1 1
4.3 odd 2 3120.2.a.w.1.1 1
5.2 odd 4 1950.2.e.g.1249.2 2
5.3 odd 4 1950.2.e.g.1249.1 2
5.4 even 2 1950.2.a.k.1.1 1
12.11 even 2 9360.2.a.p.1.1 1
13.5 odd 4 5070.2.b.d.1351.1 2
13.8 odd 4 5070.2.b.d.1351.2 2
13.12 even 2 5070.2.a.a.1.1 1
15.2 even 4 5850.2.e.e.5149.1 2
15.8 even 4 5850.2.e.e.5149.2 2
15.14 odd 2 5850.2.a.bo.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.f.1.1 1 1.1 even 1 trivial
1170.2.a.a.1.1 1 3.2 odd 2
1950.2.a.k.1.1 1 5.4 even 2
1950.2.e.g.1249.1 2 5.3 odd 4
1950.2.e.g.1249.2 2 5.2 odd 4
3120.2.a.w.1.1 1 4.3 odd 2
5070.2.a.a.1.1 1 13.12 even 2
5070.2.b.d.1351.1 2 13.5 odd 4
5070.2.b.d.1351.2 2 13.8 odd 4
5850.2.a.bo.1.1 1 15.14 odd 2
5850.2.e.e.5149.1 2 15.2 even 4
5850.2.e.e.5149.2 2 15.8 even 4
9360.2.a.p.1.1 1 12.11 even 2